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Symmetric approximate minimum degree permutation
p = symamd(S)
p = symamd(S,knobs)
[p,stats] = symamd(...)
p = symamd(S) for a symmetric positive definite matrix S, returns the permutation vector p such that S(p,p) tends to have a sparser Cholesky factor than S. To find the ordering for S, symamd constructs a matrix M such that spones(M'*M) = spones (S), and then computes p = colamd(M). The symamd function may also work well for symmetric indefinite matrices.
S must be square; only the strictly lower triangular part is referenced.
p = symamd(S,knobs) where knobs is a scalar. If S is n-by-n, rows and columns with more than knobs*n entries are removed prior to ordering, and ordered last in the output permutation p. If the knobs parameter is not present, then knobs = spparms('wh_frac').
[p,stats] = symamd(...) produces the optional vector stats that provides data about the ordering and the validity of the matrix S.
stats(1) | Number of dense or empty rows ignored by symamd |
stats(2) | Number of dense or empty columns ignored by symamd |
stats(3) | Number of garbage collections performed on the internal data structure used by symamd (roughly of size 8.4*nnz(tril(S,-1)) + 9n integers) |
stats(4) | 0 if the matrix is valid, or 1 if invalid |
stats(5) | Rightmost column index that is unsorted or contains duplicate entries, or 0 if no such column exists |
stats(6) | Last seen duplicate or out-of-order row index in the column index given by stats(5), or 0 if no such row index exists |
stats(7) | Number of duplicate and out-of-order row indices |
Although, MATLAB^{®} built-in functions generate valid sparse matrices, a user may construct an invalid sparse matrix using the MATLAB C or Fortran APIs and pass it to symamd. For this reason, symamd verifies that S is valid:
If a row index appears two or more times in the same column, symamd ignores the duplicate entries, continues processing, and provides information about the duplicate entries in stats(4:7).
If row indices in a column are out of order, symamd sorts each column of its internal copy of the matrix S (but does not repair the input matrix S), continues processing, and provides information about the out-of-order entries in stats(4:7).
If S is invalid in any other way, symamd cannot continue. It prints an error message, and returns no output arguments (p or stats).
The ordering is followed by a symmetric elimination tree post-ordering.
The authors of the code for symamd are Stefan I. Larimore and Timothy A. Davis (davis@cise.ufl.edu), University of Florida. The algorithm was developed in collaboration with John Gilbert, Xerox PARC, and Esmond Ng, Oak Ridge National Laboratory. Sparse Matrix Algorithms Research at the University of Florida: http://www.cise.ufl.edu/research/sparse/